NSC Mathematics Functions: In die diagram hieronder is die

In die diagram hieronder is die grafiek van f(x) = -2sin x die interval x ∈ [-180°; 180°] geskets - NSC Mathematics - Question 7 - 2021 - Paper 2

Question 7

In die diagram hieronder is die grafiek van f(x) = -2sin x die interval x ∈ [-180°; 180°] geskets. 7.1 Op die rooster wat in die ANTWORLDDEBOOK verskaf word, sk... show full transcript

Worked Solution & Example Answer:In die diagram hieronder is die grafiek van f(x) = -2sin x die interval x ∈ [-180°; 180°] geskets - NSC Mathematics - Question 7 - 2021 - Paper 2

Step 1

7.1 Op die rooster wat in die ANTWORDDEBOOK verskaf word, skets die grafiek van g(x) = cos(x - 60°) vir x ∈ [-180°; 180°]. Toon duidelik ALLE snitte met die asse en die draai punte van die grafiek aan.

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Answer

To sketch the graph of g(x) = cos(x - 60°) for the interval x ∈ [-180°; 180°], first determine the key features of the cosine function, including its amplitude (1), period (360°), and phase shift (60° to the right).

The turning points occur at where the cosine function reaches its maximum (1) and minimum (-1). These can be identified at:

Maximums: x = 60°, 420° (-180° is at the start of the interval, thus wrapping back)
Minimums: x = -120°

Additionally, the x-intercepts occur where g(x) = 0 depicting the points x = -30° & x = 150°. Mark these points and ensure to include a smooth transition in the sketch.

Step 2

7.2 Skryf die periode van f(3x) neer.

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Answer

The period of the function f(kx) can be calculated using the formula: $ext{Period} = \frac{360°}{k}$ For (3x) , where k = 3, the period becomes: $ext{Period} = \frac{360°}{3} = 120°.$ Thus, the period of (3x) is 120°.

Step 3

7.3 Gebruik die grafieke om die waarde van x in die interval x ∈ [-180°; 180°] te bepaal waarvoor f(x) - g(x) = 0.

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Answer

To find the values of x for which f(x) - g(x) = 0, we assess the intersecting points of both graphs obtained in the prior steps. From the sketch, the graph illustrates that the intersections are at: x = -30°, which corresponds to both functions equating at this angle within the defined interval.

Step 4

7.4 Skryf die waardeverzameling van k neer, k(x) = rac{1}{2}g(x) + 1.

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Answer

Firstly, identify the range of g(x) = cos(x - 60° which is ext{[-1, 1]}. Through transformation of the function with y = rac{1}{2}g(x) + 1 the following adjustments occur:

The amplitude transformation is halved, which changes the range to ext{[-0.5, 0.5]}.
Adding 1 shifts the entire range upward by 1, resulting in:
Range of k(x) becomes:

y ∈ [0.5, 1.5].

In die diagram hieronder is die grafiek van f(x) = -2sin x die interval x ∈ [-180°; 180°] geskets - NSC Mathematics - Question 7 - 2021 - Paper 2

Worked Solution & Example Answer:In die diagram hieronder is die grafiek van f(x) = -2sin x die interval x ∈ [-180°; 180°] geskets - NSC Mathematics - Question 7 - 2021 - Paper 2

7.1 Op die rooster wat in die ANTWORDDEBOOK verskaf word, skets die grafiek van g(x) = cos(x - 60°) vir x ∈ [-180°; 180°]. Toon duidelik ALLE snitte met die asse en die draai punte van die grafiek aan.

7.2 Skryf die periode van f(3x) neer.

7.3 Gebruik die grafieke om die waarde van x in die interval x ∈ [-180°; 180°] te bepaal waarvoor f(x) - g(x) = 0.

7.4 Skryf die waardeverzameling van k neer, k(x) = rac{1}{2}g(x) + 1.

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